April  2012, 6(2): 205-249. doi: 10.3934/jmd.2012.6.205

Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

1. 

Fakultät für Mathematik, TU Dortmund, Dortmund, Germany

2. 

CMLS École Polytechnique, Palaiseau, France

3. 

Mathematisches Institut der Ludwig-Maximilian-Universität, Munich, Germany

Received  January 2012 Published  August 2012

For a closed connected manifold $N$, we construct a family of functions on the Hamiltonian group $\mathcal{G}$ of the cotangent bundle $T^*N$, and a family of functions on the space of smooth functions with compact support on $T^*N$. These satisfy properties analogous to those of partial quasimorphisms and quasistates of Entov and Polterovich. The families are parametrized by the first real cohomology of $N$. In the case $N=\mathbb{T}^n$ the family of functions on $\mathcal{G}$ coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of $\mathcal{G}$, to Aubry--Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.
Citation: Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205
References:
[1]

A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722. doi: 10.2140/gt.2010.14.1569.  Google Scholar

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology,, Int. Math. Res. Not. IMRN, 2008 ().   Google Scholar

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A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. doi: 10.1007/BF02566074.  Google Scholar

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P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  Google Scholar

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M. Brunella, On a theorem of Sikorav, Ens. Math. (2), 37 (1991), 83-87.  Google Scholar

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M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology,, Int. Math. Res. Not., 2003 (): 1635.  doi: 10.1155/S1073792803210011.  Google Scholar

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M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.  Google Scholar

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M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.  Google Scholar

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M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), part 1, 1037-1055.  Google Scholar

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U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math., 159 (2007), 1-56. doi: 10.1007/s11856-007-0037-3.  Google Scholar

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R. Iturriaga and H. Sánchez-Morgado, A minimax selector for a class of Hamiltonians on cotangent bundles, Internat. J. Math., 11 (2000), 1147-1162.  Google Scholar

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S. Lanzat, "Symplectic Quasi-Morphisms and Quasi-States for Noncompact Symplectic Manifolds,", Ph. D. thesis, ().   Google Scholar

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S. Lanzat, Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds,, , ().   Google Scholar

[14]

R. Leclercq, Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn., 2 (2008), 249-286. doi: 10.3934/jmd.2008.2.249.  Google Scholar

[15]

J. N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems, Math. Z., 207 (1991), 169-207 doi: 10.1007/BF02571383.  Google Scholar

[16]

D. Milinković and Y.-G. Oh, Floer homology as the stable Morse homology, J. Korean Math. Soc., 34 (1997), 1065-1087.  Google Scholar

[17]

D. Milinković and Y.-G. Oh, Generating functions versus action functional. Stable Morse theory versus Floer theory, in "Geometry, Topology, and Dynamics" (Montreal, PQ, 1995), CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, (1998), 107-125.  Google Scholar

[18]

A. Monzner and F. Zapolsky, A comparison of symplectic homogenization and Calabi quasi-states, J. Topol. Anal., 3 (2011), 243-263. doi: 10.1142/S1793525311000581.  Google Scholar

[19]

Y.-G. Oh, Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Diff. Geom., 46 (1997), 499-577.  Google Scholar

[20]

Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants, Comm. Anal. Geom., 7 (1999), 1-54.  Google Scholar

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Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in "The Breadth of Symplectic and Poisson Geometry," Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 525-570.  Google Scholar

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G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Mosc. Math. J., 3 (2003), 593-619, 745.  Google Scholar

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L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphisms," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.  Google Scholar

[24]

L. Polterovich and K. F. Siburg, On the asymptotic geometry of area-preserving maps, Math. Res. Lett., 7 (2000), 233-243.  Google Scholar

[25]

P. Py, Quelques plats pour la métrique de Hofer, J. Reine Angew. Math., 620 (2008), 185-193. doi: 10.1515/CRELLE.2008.053.  Google Scholar

[26]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.  Google Scholar

[27]

K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 92 (1998), 295-319. doi: 10.1215/S0012-7094-98-09207-9.  Google Scholar

[28]

K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics," Lecture Notes in Mathematics, 1844, Springer-Verlag, Berlin, 2004. doi: 10.1007/b97327.  Google Scholar

[29]

A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403. doi: 10.2140/gt.2010.14.2383.  Google Scholar

[30]

D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology Appl., 96 (1999), 249-266. doi: 10.1016/S0166-8641(98)00049-2.  Google Scholar

[31]

Viterbo, C., Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685-710. doi: 10.1007/BF01444643.  Google Scholar

[32]

C. Viterbo, Symplectic homogenization,, , ().   Google Scholar

[33]

F. Zapolsky, On the Lagrangian Hofer geometry in symplectically aspherical manifolds,, , ().   Google Scholar

show all references

References:
[1]

A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722. doi: 10.2140/gt.2010.14.1569.  Google Scholar

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology,, Int. Math. Res. Not. IMRN, 2008 ().   Google Scholar

[3]

A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., 53 (1978), 174-227. doi: 10.1007/BF02566074.  Google Scholar

[4]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420.  Google Scholar

[5]

M. Brunella, On a theorem of Sikorav, Ens. Math. (2), 37 (1991), 83-87.  Google Scholar

[6]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology,, Int. Math. Res. Not., 2003 (): 1635.  doi: 10.1155/S1073792803210011.  Google Scholar

[7]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43.  Google Scholar

[8]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: 10.1112/S0010437X0900400X.  Google Scholar

[9]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), part 1, 1037-1055.  Google Scholar

[10]

U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math., 159 (2007), 1-56. doi: 10.1007/s11856-007-0037-3.  Google Scholar

[11]

R. Iturriaga and H. Sánchez-Morgado, A minimax selector for a class of Hamiltonians on cotangent bundles, Internat. J. Math., 11 (2000), 1147-1162.  Google Scholar

[12]

S. Lanzat, "Symplectic Quasi-Morphisms and Quasi-States for Noncompact Symplectic Manifolds,", Ph. D. thesis, ().   Google Scholar

[13]

S. Lanzat, Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds,, , ().   Google Scholar

[14]

R. Leclercq, Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn., 2 (2008), 249-286. doi: 10.3934/jmd.2008.2.249.  Google Scholar

[15]

J. N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems, Math. Z., 207 (1991), 169-207 doi: 10.1007/BF02571383.  Google Scholar

[16]

D. Milinković and Y.-G. Oh, Floer homology as the stable Morse homology, J. Korean Math. Soc., 34 (1997), 1065-1087.  Google Scholar

[17]

D. Milinković and Y.-G. Oh, Generating functions versus action functional. Stable Morse theory versus Floer theory, in "Geometry, Topology, and Dynamics" (Montreal, PQ, 1995), CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, (1998), 107-125.  Google Scholar

[18]

A. Monzner and F. Zapolsky, A comparison of symplectic homogenization and Calabi quasi-states, J. Topol. Anal., 3 (2011), 243-263. doi: 10.1142/S1793525311000581.  Google Scholar

[19]

Y.-G. Oh, Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Diff. Geom., 46 (1997), 499-577.  Google Scholar

[20]

Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants, Comm. Anal. Geom., 7 (1999), 1-54.  Google Scholar

[21]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in "The Breadth of Symplectic and Poisson Geometry," Progr. Math., 232, Birkhäuser Boston, Boston, MA, (2005), 525-570.  Google Scholar

[22]

G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Mosc. Math. J., 3 (2003), 593-619, 745.  Google Scholar

[23]

L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphisms," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.  Google Scholar

[24]

L. Polterovich and K. F. Siburg, On the asymptotic geometry of area-preserving maps, Math. Res. Lett., 7 (2000), 233-243.  Google Scholar

[25]

P. Py, Quelques plats pour la métrique de Hofer, J. Reine Angew. Math., 620 (2008), 185-193. doi: 10.1515/CRELLE.2008.053.  Google Scholar

[26]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419-461. doi: 10.2140/pjm.2000.193.419.  Google Scholar

[27]

K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 92 (1998), 295-319. doi: 10.1215/S0012-7094-98-09207-9.  Google Scholar

[28]

K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics," Lecture Notes in Mathematics, 1844, Springer-Verlag, Berlin, 2004. doi: 10.1007/b97327.  Google Scholar

[29]

A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403. doi: 10.2140/gt.2010.14.2383.  Google Scholar

[30]

D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology Appl., 96 (1999), 249-266. doi: 10.1016/S0166-8641(98)00049-2.  Google Scholar

[31]

Viterbo, C., Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), no. 4, 685-710. doi: 10.1007/BF01444643.  Google Scholar

[32]

C. Viterbo, Symplectic homogenization,, , ().   Google Scholar

[33]

F. Zapolsky, On the Lagrangian Hofer geometry in symplectically aspherical manifolds,, , ().   Google Scholar

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